Aposyndesis in $\mathbb {N}$

José del Carmen Alberto-Domínguez; Gerardo Acosta; Maira Madriz-Mendoza

Displaying similar documents to “Aposyndesis in $ℕ$ ”

Some properties of the class of arithmetic functions $T_{r} (N)$

R. P. Pakshirajan (1963)

Annales Polonici Mathematici

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On integers of the form $p + 2^{k}$

Laurent Habsieger, Xavier-François Roblot (2006)

Acta Arithmetica

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On a certain class of arithmetic functions

Antonio M. Oller-Marcén (2017)

Mathematica Bohemica

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A homothetic arithmetic function of ratio $K$ is a function $f : ℕ \to R$ such that $f (K n) = f (n)$ for every $n \in ℕ$ . Periodic arithmetic funtions are always homothetic, while the converse is not true in general. In this paper we study homothetic and periodic arithmetic functions. In particular we give an upper bound for the number of elements of $f (ℕ)$ in terms of the period and the ratio of $f$ .

Numerical characterization of nef arithmetic divisors on arithmetic surfaces

Atsushi Moriwaki (2014)

Annales de la faculté des sciences de Toulouse Mathématiques

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In this paper, we give a numerical characterization of nef arithmetic $ℝ$ -Cartier divisors of $C^{0}$ -type on an arithmetic surface. Namely an arithmetic $ℝ$ -Cartier divisor $\bar{D}$ of $C^{0}$ -type is nef if and only if $\bar{D}$ is pseudo-effective and $\hat{\deg} ({\bar{D}}^{2}) = \hat{vol} (\bar{D})$ .

On generalized square-full numbers in an arithmetic progression

Angkana Sripayap, Pattira Ruengsinsub, Teerapat Srichan (2022)

Czechoslovak Mathematical Journal

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Let $a$ and $b \in ℕ$ . Denote by $R_{a, b}$ the set of all integers $n > 1$ whose canonical prime representation $n = p_{1}^{α_{1}} p_{2}^{α_{2}} \dots p_{r}^{α_{r}}$ has all exponents $α_{i}$ $(1 \leq i \leq r)$ being a multiple of $a$ or belonging to the arithmetic progression $a t + b$ , $t \in ℕ_{0} : = ℕ \cup {0}$ . All integers in $R_{a, b}$ are called generalized square-full integers. Using the exponent pair method, an upper bound for character sums over generalized square-full integers is derived. An application on the distribution of generalized square-full integers in an arithmetic progression is given. ...

A structure theorem for sets of small popular doubling

Przemysław Mazur (2015)

Acta Arithmetica

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We prove that every set A ⊂ ℤ satisfying $\sum_{x} m i n (1_{A} * 1_{A} (x), t) \leq (2 + δ) t | A |$ for t and δ in suitable ranges must be very close to an arithmetic progression. We use this result to improve the estimates of Green and Morris for the probability that a random subset A ⊂ ℕ satisfies |ℕ∖(A+A)| ≥ k; specifically, we show that $ℙ (| ℕ ∖ (A + A) | \geq k) = Θ (2^{- k / 2})$ .

On arithmetic progressions on Edwards curves

Enrique González-Jiménez (2015)

Acta Arithmetica

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Let $m \in ℤ_{> 0}$ and a,q ∈ ℚ. Denote by $_{m} (a, q)$ the set of rational numbers d such that a, a + q, ..., a + (m-1)q form an arithmetic progression in the Edwards curve $E_{d} : x ² + y ² = 1 + d x ² y ²$ . We study the set $_{m} (a, q)$ and we parametrize it by the rational points of an algebraic curve.

Generalized weighted quasi-arithmetic means and the Kolmogorov-Nagumo theorem

Janusz Matkowski (2013)

Colloquium Mathematicae

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A generalization of the weighted quasi-arithmetic mean generated by continuous and increasing (decreasing) functions $f ₁, . . ., f_{k} : I \to ℝ$ , k ≥ 2, denoted by $A^{[f ₁, . . ., f_{k}]}$ , is considered. Some properties of $A^{[f ₁, . . ., f_{k}]}$ , including “associativity” assumed in the Kolmogorov-Nagumo theorem, are shown. Convex and affine functions involving this type of means are considered. Invariance of a quasi-arithmetic mean with respect to a special mean-type mapping built of generalized means is applied in solving a functional equation. For...

An inconsistency equation involving means

Roman Ger, Tomasz Kochanek (2009)

Colloquium Mathematicae

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We show that any quasi-arithmetic mean $A_{φ}$ and any non-quasi-arithmetic mean M (reasonably regular) are inconsistent in the sense that the only solutions f of both equations $f (M (x, y)) = A_{φ} (f (x), f (y))$ and $f (A_{φ} (x, y)) = M (f (x), f (y))$ are the constant ones.

A problem of Rankin on sets without geometric progressions

Melvyn B. Nathanson, Kevin O'Bryant (2015)

Acta Arithmetica

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A geometric progression of length k and integer ratio is a set of numbers of the form $a, a r, . . ., a r^{k - 1}$ for some positive real number a and integer r ≥ 2. For each integer k ≥ 3, a greedy algorithm is used to construct a strictly decreasing sequence ${(a_{i})}_{i = 1}^{\infty}$ of positive real numbers with a₁ = 1 such that the set $G^{(k)} = ⋃_{i = 1}^{\infty} (a_{2 i}, a_{2 i - 1}]$ contains no geometric progression of length k and integer ratio. Moreover, $G^{(k)}$ is a maximal subset of (0,1] that contains no geometric progression of length k and integer ratio. It is also proved that...

Precompactness in the uniform ergodic theory

Yu. Lyubich, J. Zemánek (1994)

Studia Mathematica

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We characterize the Banach space operators T whose arithmetic means ${n^{- 1} (I + T + . . . + T^{n - 1})}_{n \geq 1}$ form a precompact set in the operator norm topology. This occurs if and only if the sequence ${n^{- 1} T^{n}}_{n \geq 1}$ is precompact and the point 1 is at most a simple pole of the resolvent of T. Equivalent geometric conditions are also obtained.

General position properties in fiberwise geometric topology

Taras Banakh, Vesko Valov

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General position properties play a crucial role in geometric and infinite-dimensional topologies. Often such properties provide convenient tools for establishing various universality results. One of well-known general position properties is DDⁿ, the property of disjoint n-cells. Each Polish $L C^{n - 1}$ -space X possessing DDⁿ contains a topological copy of each n-dimensional compact metric space. This fact implies, in particular, the classical Lefschetz-Menger-Nöbeling-Pontryagin-Tolstova embedding...

Algebraic and topological structures on the set of mean functions and generalization of the AGM mean

Bakir Farhi (2013)

Colloquium Mathematicae

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We present new structures and results on the set of mean functions on a given symmetric domain in ℝ². First, we construct on a structure of abelian group in which the neutral element is the arithmetic mean; then we study some symmetries in that group. Next, we construct on a structure of metric space under which is the closed ball with center the arithmetic mean and radius 1/2. We show in particular that the geometric and harmonic means lie on the boundary of . Finally, we give...

A functional equation involving f and $f^{- 1}$ .

J. S. Ratti, Y. -F. Lin (1990)

Colloquium Mathematicae

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Goldbach’s problem with primes in arithmetic progressions and in short intervals

Karin Halupczok (2013)

Journal de Théorie des Nombres de Bordeaux

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Some mean value theorems in the style of Bombieri-Vinogradov’s theorem are discussed. They concern binary and ternary additive problems with primes in arithmetic progressions and short intervals. Nontrivial estimates for some of these mean values are given. As application inter alia, we show that for large odd $n \neg \equiv 1 (6)$ , Goldbach’s ternary problem $n = p_{1} + p_{2} + p_{3}$ is solvable with primes $p_{1}, p_{2}$ in short intervals $p_{i} \in [X_{i}, X_{i} + Y]$ with $X_{i}^{θ_{i}} = Y$ , $i = 1, 2$ , and $θ_{1}, θ_{2} \geq 0.933$ such that $(p_{1} + 2) (p_{2} + 2)$ has at most $9$ prime factors.

Density of solutions to quadratic congruences

Neha Prabhu (2017)

Czechoslovak Mathematical Journal

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A classical result in number theory is Dirichlet’s theorem on the density of primes in an arithmetic progression. We prove a similar result for numbers with exactly $k$ prime factors for $k > 1$ . Building upon a proof by E. M. Wright in 1954, we compute the natural density of such numbers where each prime satisfies a congruence condition. As an application, we obtain the density of squarefree $n \leq x$ with $k$ prime factors such that a fixed quadratic equation has exactly $2^{k}$ solutions modulo $n$ . ...

Iterated quasi-arithmetic mean-type mappings

Paweł Pasteczka (2016)

Colloquium Mathematicae

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We work with a fixed N-tuple of quasi-arithmetic means $M ₁, . . ., M_{N}$ generated by an N-tuple of continuous monotone functions $f ₁, . . ., f_{N} : I \to ℝ$ (I an interval) satisfying certain regularity conditions. It is known [initially Gauss, later Gustin, Borwein, Toader, Lehmer, Schoenberg, Foster, Philips et al.] that the iterations of the mapping $I^{N} ∋ b \mapsto (M ₁ (b), . . ., M_{N} (b))$ tend pointwise to a mapping having values on the diagonal of $I^{N}$ . Each of [all equal] coordinates of the limit is a new mean, called the Gaussian product of the means $M ₁, . . ., M_{N}$ taken...

Displaying similar documents to “Aposyndesis in ℕ ”

Displaying similar documents to “Aposyndesis in $ℕ$ ”